"Or not" doesn't really seem to be needed to complement "whether". Why do people use it then? Is it redundancy and nothing more? Or is it for emphasis? Or are there cases when "or not" is required for the sentence to be grammatical?

As Henry Higgins observed in Pygmallion, the best grammarians are often those who learned English in school as immigrants. My parents, who were first-generation Americans in the early 20th century, learned English grammar in NYC public schools meticulously. They insisted "whether or not" is proper usage, period. Over time, language evolves or erodes and the rules change, which really means there are no authorities. I believe many changes are driven by relatively poorly educated TV personalities misusing words, which then become common usage. Someone once said, "C students rule the world."
–
user52936Sep 27 '13 at 13:03

The "If/*whether you decide to come, give me a call." example given in an answer to a strongly related question makes me question this pronouncement. I would hope some logical or grammatical rule would prevent the "Whether you decide to come, give me a call." option in such a case, even though I can't cite a specific rule myself.
–
John ToblerSep 19 '11 at 22:36

Could you give me your thoughts on my answer? I'd be interested in hearing what you have to say about it.
–
timothymhJul 2 '12 at 2:34

@JohnT I think whether emphasizes a dichotomy in which one option is imminent, though the option may not be known. If emphasizes the conditional nature of the matter.
–
fredsbendMar 3 '14 at 16:00

Although perhaps implied in the answers above, I think it could be clarified a bit: Whether is binary. In a sentence where two options are offered, no 'or not' is required - in fact it would be quite cumbersome. However, if the option is to do or not to do, is or isn't, then the 'or not' is required.

I don't know whether I should have the fish or the chicken.
I don't know whether I should eat or not.

I'm surprised nobody mentioned this that I saw, so I'll post an answer.

The word whether should be used by itself in the situations you mentioned above. The “or not” is a mistaken crossover from the correct usage of “whether or not” mentioned below. It is often used that way, but when writing it's best to avoid that unnecessary bit.

The phrase whether or not is a condition, used in statements to show that something will or will not happen, regardless of certain other variables:

I don't agree that the addition of or not is a "mistaken crossover". Whether introduces two or more possibilities; adding or not is merely clarifying what one of the possibilities is, whether or not it's obvious. ;-)
–
JezJul 2 '12 at 10:12

It's worth pointing out that, etymologically speaking, the roots of whether are which/either of two. It's inherently a "binary choice" word, so whereas "I don't know whether it be fish or fowl" is fine, "I don't know whether it be fish or fowl or good red herring" isn't really grammatical. Which is not to say people never use that extended form - but it does sometimes attract criticism.

With "unary choice" forms such as "I don't know whether I like it", the alternative ("I don't like it") can invariably be shortened to "or not" - or simply discarded completely, since it's implicit anyway.

Possibly some will say if only one choice is presented, you should use "if" rather than "whether", but skimming through written instances of "Tell me whether" suggests that most people have always been quite relaxed on that point.

TL;DR: "or not" is never required if the alternative is a simple negation of the stated proposition, but an "or" clause is required in, say, "You must choose whether to write novels or poems" (presupposing that writing software is not an option currently on offer).